CURRENCY INVARIANCE, OPTIMAL CURRENCY BASKETS, AND SYNTHETIC DOLLARS1 Nikolai V. Hovanov, James W. K

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CURRENCY INVARIANCE, OPTIMAL CURRENCY BASKETS,
AND SYNTHETIC DOLLARS1 Nikolai V. Hovanov,
James W. Kolari, and Mikhail V. Sokolov St.
Petersburg State University, Texas AM
University, and A.V.K. Investment Company,
respectively Prepared for the Inaugural
Conference (Nul-Lustrum) Utrecht School of
EconomicsOctober 21-23, 2003Utrecht University,
Netherlands1 The authors gratefully
acknowledge financial support from the Center for
International Business Studies in the Mays
Business School at Texas AM University.
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This Paper

• Review currency invariant index and optimal
  currency basket index concepts in Hovanov,
  Kolari, and Sokolov (forthcoming paper).
• Discuss practical implications of stable
  aggregate currency (SAC) indexes.
• Present a new optimal currency basket using
  methods in Hovanov et al. namely, a synthetic
  dollar, defined as a currency basket containing
  no dollars but closely tracking the U.S. dollar
  over time.

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Currency Invariance

• Hovanov et al. proposed the concept of a currency
  invariant index

This index is the same no matter what base
currency i is used. Example NVal(USD)
EURO/USD / (EURO/USD x EURO/YEN x EURO/EURO)1/3
YEN/USD / (YEN/USD x YEN/YEN x
YEN/EURO)1/3 USD/USD / (USD/USD x USD/YEN
x USD/EURO)1/3 NVal(USD) (USD/USD x YEN/USD x
EURO/USD)1/3.
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Effective Exchange Rates

• Definition of EER with No Trade Weights (Equal
  Weights)
• EER(USD) (YEN/USD x EURO/USD)1/2
• Note! This value is different from our currency
  invariant index
• NVal(USD) (USD/USD x YEN/USD x EURO/USD)1/3

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Effective Exchange Rates

• Trade Weights if we insert trade weights, as in
  conventional EERs, then currency invariance is
  violated.
• EERs will vary depending on the base currency
  that you pick.
• EER(USD) (w1 YEN/USD x w2 EURO/USD)1/2
• (YEN/USD) / (w3 YEN/EURO x w1 YEN/USD x
  YEN/YEN)1/2
• (EURO/USD) / (1/w3 EURO/YEN x w2 EURO/USD x
  EURO/EURO)1/2
• (USD/USD) / (1/w1 USD/YEN x 1/w2 USD/EURO x
  USD/USD)1/2
• For currency invariance to hold, these
  alternative forms of EER(USD) would have to be
  equal to one another. This will not be true, as
  the weights are not selected to maintain currency
  invariance.

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Why Currency Invariance?

• Whenever base currency choice results in
  ambiguous results, currency invariant indexes
  could be helpful.
• Hovanov et al. sought a minimum variance currency
  basket. However, depending on the base currency
  that is used, the composition of this optimal
  currency basket will change. So, given over 150
  currencies in the world that could be used as a
  base currency, there are many possible optimal
  currency baskets.
• Currency invariant indexes allow single values
  for each currency regardless of base currency
  choice. Now only one optimal currency basket is
  obtained.

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Symmetry Assumption in Currency Invariance

• The equal weights on the currencies in the
  geometric mean arises due to an assumption of
  symmetry.
• This assumption implies that currencies are
  perfect substitutes for one another (e.g., the
  well-known expectations theory of exchange rates
  makes this same assumption).
• For soft currencies this assumption may well not
  hold. Thus, currency invariant indexes are most
  appropriately applied to hard currencies of major
  industrial countries.

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Optimal Currency Baskets
Goal We want to find the optimal weights w
that will minimize the variance of Ind(wt)
,

This problem can be solved by using quadratic
linear programs like those used in Markowitz
mean-variance analyses (e.g., Newtons method
based on the application Solver.xls in MS Excel
7.0). However, unlike mean-variance analyses, we
are interested in the global minimum variance,
rather than conditional minimum variance given
some mean return.
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Stable Aggregate Currency (SAC)

• We have collected daily exchange rate data from
  January 1, 1989 to December 31, 1998 (prior to
  the euros introduction) for the British pound
  sterling (GBP), French franc (FRF), German mark
  (DEM), Japanese yen (JPY), and U.S. dollar (USD).
  

Here we see that SAC has very low volatility
more than 25 times smaller standard deviation
than the French franc (where values of currencies
are computed using currency invariant indexes).
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Applications of SAC

• Numeraire for Currency Valuation. As a currency
  index, SAC can be used as a benchmark to measure
  the value of currencies over time. This
  application is similar in spirit to using the
  International Monetary Funds Special Drawing
  Right (SDR), the U.S. Federal Reserves Major
  Currency Index, etc. However, SAC is a more
  stable base.
• For example, if the USD/SDR increased, was this
  increase attributable to an increase in the U.S.
  dollar or a decrease in the basket value of SDR?
  This problem is made worse by using individual
  base currencies. If the USD/EUR exchange rate
  increased, was it due to an increase in the U.S.
  dollar or a decrease in the European euro? SAC
  mitigates the problem of movement in the base
  currency over time. How is a countrys local
  currency changing in world currency markets over
  the past week, month, year, etc.? SAC provides a
  stable base currency basket against which to
  measure currency values over time.

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Applications of SAC

• Denomination of Asset Prices and Global Returns.
  Suppose that we want to compute the rate of
  return on Microsofts common stock over the past
  year. Of course, since Microsoft is purchased by
  investors in many countries using different local
  currencies, there are many rates of return series
  for this stock. Each rate of return will differ
  due to different base currency choice. Using the
  SAC/USD exchange rate, we can convert dollar
  prices of Microsoft to SAC prices, and then
  compute its rate of return in terms of SAC. This
  rate of return would be common to all investors
  around the world. Any local currency return
  would equal the SAC-based return plus the return
  on the local currency per unit of SAC. We will
  refer to the common SAC-based return as the
  global rate of return.
• Our global return concept is analogous to the
  real rate of return, and the return on local
  currencies per unit SAC is similar to the
  inflation rate. If inflation increased 10
  during the year, and Microsofts rate of return
  in nominal terms increased 10, the real rate of
  return was zero. Following this logic, if the
  dollar increased by 10 against the SAC over the
  past year, and Microsofts rate of return
  increased by 10, the stocks global rate of
  return would be zero.

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Applications of SAC

• Cash Settlement Index on Debt Contracts.
  Related to the denomination of asset prices, SAC
  could be used as a cash settlement index on debt
  contracts. For example, suppose that a country
  issues debt into world markets and denominates
  its debt in U.S. dollars. If the dollar
  increases in value in world currency markets,
  this currency movement will increase the nations
  cost of debt. Alternatively, if interest and
  principal are denominated in SAC, this problem
  would be greatly reduced, as SAC is quite stable
  over time.

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Applications of SAC

• World Money Units. The use of SAC as a form of
  world money in line with Mundells ideas.
  Businesses in different countries could make
  payments to one another using SAC. A bank could
  serve as intermediary to handle transfers of
  multi-currency baskets of funds. Account
  balances would reflect not only the number of
  world money units but the quantities of
  individual currencies. Banks could charge a
  service fee for this derivative security
  business. Naturally, businesses making and
  receiving payments in SAC have no currency or
  exchange rate risk.
• However, if a business sought to convert its
  SAC holdings to a local currency, there would be
  currency risk. A question arises here on the
  relative currency risk of SAC/local currency
  versus say U.S. dollars/local currency or some
  other exchange rate. How stable is the SAC/local
  currency exchange rate compared to exchange rates
  using local currencies in both the numerator and
  denominator. This comparison is an empirical
  issue for future study.

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Synthetic Dollars

• Use currency invariant indexes for currency
  values.
• Find the currency basket Ind with maximum
  correlation with the U.S. dollar

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Synthetic Dollar Versus U.S. Dollar (Using
Currency Invariant Indexes) Daily 2002 Data
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Applications of Synthetic Money

• Countries that peg to the dollar but for
  political, social, etc. reasons would prefer to
  use synthetic dollars as a peg. China and other
  Asian countries are possible users. A soft peg
  could be developed by constructing a
  quasi-synthetic dollar that only had (say) a
  correlation coefficient equal to 0.70 to the U.S.
  dollar.
• Some large firms issuing global bonds in many
  countries may prefer to denominate some of their
  bonds in synthetic dollars. Some investors may
  prefer to avoid dollar-denominated bonds.
• A series that mimics the old French franc,
  German mark, etc. could be constructed and used
  for payments, debt contracts
• Other applications?

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Summary/Conclusions

• Currency invariant indexes help to solve problems
  with base currency choice.
• SAC has numerous practical applications,
  especially as a currency index (e.g., it is a
  stable currency basket to use as a numeraire in
  valuing individual currencies).
• Synthetic money can be created using currency
  invariant indexes and minimization analyses
  similar to solving for SAC.